which is the zero trivector (i.e., a degenerate plane formed by wedging together three collinear points), so the point does indeed lie on the line!

I'd also like to mention that while writing the calculations out by hand appears tedious (and it is), an efficient computer implementation of the above only performs 12 floating-point muls and 8 floating-point adds.

The ordering of basis blade indices, handling which coefficients go where, etc. can be performed at compile-time with template meta-programming techniques in the C++ or D programming languages.

Storage requirements are fairly undemanding: for 3-d k-flats, we need

4 floats to store a point,

4 floats for a plane, and

6 floats for a line.

This is pretty much matches the storage requirements for these same entities within a standard 3-d engine setting. Note, however, that a line doesn't directly store any information about the points used to construct it (it's not a line segment).